LINEAR DIFFERENCE AND DIFFERENTIAL EQUATIONS
SATISFYING CONDITIONS AT MORE THAN
ONE POINT1
TOMLINSON FORT
1. Introduction.
The problem of differential equations satisfying
conditions at more than one point is not new. W. M. Whyburn2 has
given an extensive bibliography
to which the reader is referred. In
the present paper sufficient conditions on the coefficients are given
that a solution of a system of linear equations exist consisting of a
set of functions some of which take on prescribed values at one point
and others at other points. The reader is referred to Theorems II
and I for precise statements. The approach to differential equations is
through difference equations. It is realized that most mathematicians
will probably regard the differential
equations as the more interesting. However, the author regards difference equations as of interest in themselves and Theorem I as of equal interest with Theorem II.
In a way it is more fundamental
since, as in many other places, the
facts for the differential equations are inferred from those for difference equations.
2. The difference equation. Consider the system of difference equations
n
(1)
Here i is limited
fined when
y,(i + 1) = 2~1<^n(i)y^(i),
/i=i
to integral
(2)
There is no gain in generality
values and the coefficients
v = 1, ■■■,n.
a„„(i) are de-
0 ^ i < b.
if (2) is replaced
by
agi<b.
By a solution of (1) we shall mean a set of functions y,(i) which
satisfy (1) at all points of (2). It is well known and immediately
proved that there exists one and only one solution of (1) having arbitrary prescribed values at 0.
Presented to the Society August 27, 1957; received by the editors June 30, 1957
and, in revised form, September 3, 1957.
1 This research was supported by the United States Air Force through the Air
Force Office of Scientific Research of the Air Research and Development Command
under contract No. AF 18(603-23).
8 Proceedings of the conference on differential equations, edited by L. B. Diaz and
L. B. Payne, University of Maryland Bookstore, 1956, p. 1.
287
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288
TOMLINSONFORT
[April
Suppose we are given an integer a, where 0 = a <b. Begin with yv(a)
and calculate successively yi<(a + l), yv(a + 2), • • ■, yv (k) by means of
(1), v = l,2,
■ • • ,n. Then let
n
(3)
y,(k) = E A,a(a, k)ya(a),
c=i
(4)
D,(k, a, fi) = det. aapBq(k),
p, q = 1, • • • , v,
(5)
H,(a, k, a, fi) = det. Aajq(a, k),
p, q = 1, • • • , v,
m
(6)
(at, 8s, fi, y, m) = E <*«ti3„(*)^Wa>k)>
n-1
in
(6')
(at, Se,m) = E ^(^^(a,
A),
H=l
(7)
A,(a, ^, a, /3, y, S) = det. («„, Sq,fi,y,v),
p, q = 1, ■ • • , v
= D,(k,a,fi)H,(a,k,y,S).
Lemma. Hypothesis.
ai<a2<
■ ■ ■ <av,fii<fi2<
Conclusion
Dv(k, a, fi)>0
• ■ ■ <fiv.
whenever
a = k<c
and
:
(8) H,(a, c, y, 8) > 0 whenever 71 < 72 < • • • < y„ S\ < 82 < • ■ • < 5„.
Proof is by means of mathematical
induction.
To begin with
H,(a, a + 1, a, fi) = D,(a, a, fi) > 0.
This is the first step in the induction.
We shall next prove that
(9)
H,(a,k+l,a,fi)=
V
Dy(k, a, v)Hv(a, k, v, fi)
11. • • -.i»
where
the
summation
Iii fii * • • »ij» subject
is taken
for all possible
to the restrictions
Proof is by straightforward
r]i<r]t<
combinations
■ • • <r\v.
verification.
From (3) we have
n
yr(k + 1) = E
A,u(a, k + l)yM(a).
K-l
Moreover
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of
i958]
DIFFERENCE AND DIFFERENTIAL EQUATIONS
289
V
yv(k + 1) = 2~2a„„(k)yu(k)
x-i
n
n
= 2 av»(k)H 4„„(a, k)yp(a)
H=l
p=l
n
n
= Jl2~l a^fyA^a, k)y„(a)
n=i
n
P=i
r~
n
= Z)
£
—I
a^(k)A„D(a, k) yp(a).
P=i L ^=i
Equating
J
coefficients we have
n
(10)
Av„(a, k + 1) = 2~Lam(k)Aup(a, k) = (v, p, n).
e=i
As a consequence
(11)
of this and (5)
Ht(a, k + 1, a, p) = det. (ap, Pq, n),
p, q = 1, • • • , n.
Note that although the determinant
is of order v each element is
the sum of n terms. We expand this determinant
into n" determinants
by columns thus
(12)
Hr(a, k + 1, a, p) =
X) det. aapVq(k)AVq8q(a,
k),
ii. ••••ip
P,q=l,---,v
Here the summation
is extended to all possible combinations
of
771,772,■ ■ • , J?*,subject only to the restrictions 0<r)j^n,
j = 1, • • • , v.
We denote the determinants
in (12) by
(13)
»(,!,
• • • , r,r).
We note that if any two 77's are the same SD= 0. This is true since 3D
is then a determinant
with two columns proportional.
There remain
n(n —1) • • • (n—v + l)=P.
determinants.
We denote
these by
On • • • i 2Dp-In general O.-^O,- if i?*j. We note from (7) that
(14)
2
Dv(k,a,p)Hr(a,k,r,,p)=
£
Ay(a, k, a, V, V, p).
We now expand all determinants
in the right hand member of (14)
also by columns. Each of the determinants
2Di, • • ■ , SDpoccurs once
and only once. Other determinants
which occur in the expansion are
zero having two columns proportional
as explained above. We thus
establish formula (9). Induction is now immediate and the lemma is
proved.
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290
TOMLINSON FORT
Theorem
I. Let 0 = &0<&i< • ■ ■ <kq-i^b
= n be integers.
D,(k,
a,
[April
and 0<pi<
Let Gi, • • • , Gn be arbitrary
fi)>0
whenever ax<a2<
■ ■ • <a„,
• • ■ <pq
numbers.
Suppose
fii<fi2<
■ ■ • <fi„
kj-i = k<kj and v = n —pj, j = l, • • • , q —l. Then there exists one and
only one solution of (I) such that
yPi+i(kj) = Gpj+i,
* = li • • • i Pi+i - Pitj = o, • • •, q - 1, Po = 0, &o= o.
To prove this we first fix yi(0)=Gi,
■ • ■ , yn(kq)=Gn.
With this
done we shall show that under the hypotheses
of the theorem it is
possible to determine yi(0), y2(0), • • • , y„(0) in one and only one
way, hence to determine
one and only one solution with the prescribed values.
We write down (3) for v = pi + l, ■ • • , n and k = ki. We know by
the lemma that Hn-P1(0, ki, a, fi)>0.
Hence we can solve for
ypl+i, (0), • • • , y„(0) in terms of yPl+i(ki), • ■ ■ , yn(ki). We note that
yvi+i(ki), • • • , yP2(ki) have known values namely GPl+i, • • ■ , GPr
We write
(15)
y„(0) = /,
[Gu ■ ■ ■, GP2,yP2+i(ki), ■ ■ ■, yn(ki)],
v = 1, • • •, n.
We now write (3) for k = kt, a = ki, v = pt + l, ■ ■ ■ , n. We solve these
equations for yV2+i(ki), ■ ■ ■ , yn(ki) in terms of yP2+i(&2), ■ • ■ ,yn(kt).
We can do this since iT„_Ps(&i, kt, a, fi)>0 by the lemma. We substitute these values in (15) noting that
y3>2+i(&2) =
GP2+i,
• • • , yP3 =
GPi.
We write
(2)
(16)
y„(0) = /,
We solve
[Gi, • • • , GPl, yP3+i(k2), • • • , yn(k2)],
for yP3+i(&2), • • • , yn(k2) in terms
and substitute
y,(0) = /,
We continue
v = 1, • • • , n.
of ypa+i(&3), • • • , yn(k3)
in (16) getting
[Gi, ■ • ■ , GPi, yPi+i(ki), • • ■ , yn(k3)],
v = 1, • • • , n.
this process until we finally have
yr(0)=/,(*_1)[Gi,
and the theorem
••-,£„]
is proved.
3. The differential
tions
equation.
Consider the set of differential
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equa-
1958]
DIFFERENCE AND DIFFERENTIAL EQUATIONS
(17)
where
dv
."
dx
,^=1
— = 52 gw(x)y>,
291
v = 1, ■■■,n
gru(x) are continuous.
0 ^ x g 1.
Assume
/?>0.
Let i?„„(x) =l+fegw(x)
and RVil(x) =hgvli(x), VT^p.. Let
£>„(x; a, P) = det. J?arg„
and assume
Theorem
ai<a2<
• • • <ar;
Pi<P2<
r, s = 1, • • • , v
■ ■ • <P,.
II. Let 0 = fc0<&i< • ■ ■ <kt£l.
Let 0<pi<p2<
• ■■
<pq = n be integers and let Gi, G2, • ■ ■ , Gn be arbitrary numbers. Suppose for sufficiently small h Dn~Pi(x, a, P)>0
when k}-i^x<kj,
j=l,
■ ■ • , n. Then there exists one and only one solution of (17) such
that
yPj+i = Gpf+i]
po = 0, ko = 0,
i = 1, ■■■, pj+i - pj]j = 0, ■ • • , q - 1.
Proof will be made to depend upon the corresponding
difference equations.
Assume x, so chosen that Xi/h = i a positive integer
theorem
for
or zero. Let
K(i) = g,„(hi) = g>u(xi).
Consider
the difference
equations
1
"
—Ayv(i)=Jlbm(i)yu(i),
h
„=i
(18)
We write
these equations
v = 1, - - • , n.
in the form
n
yy(i + 1) = X) <*m(i)%(i),
v = 1, • • • , n
n-i
where
avu(i) =Rva(xi)
that
is aPli(i) =hbvu(i),
lit^v
and
a„(i) = 1 + hb„(i).
We expect to consider (18) when h approaches
0. In place of hi we
have written x,-. We also write yr(i) or jv(xi) indiscriminately
meaning
exactly the same thing in each instance. Let h be small. Mark the
points (x^ y,(xi)) in the Cartesian plane and connect them by straight
line segments. We then write y,(x) for the function defined by this
graph. It is well known3 that if a solution of (17) satisfying conditions
3 See, for example,
Fort,
Finite differences and difference equations in the real
domain, Clarendon Press, Oxford, 1948, p. 164.
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292
TOMLINSON FORT
yi(0) =Ci, y2(0) =c2, ■ ■ ■ , yn(0) =cn is given and if a solution
of (18)
is determined
satisfying
yi(0) =Ci(h), • ■ ■ , yn(0)=cn(h)
and
lim/,,o cv(h) =cv, v = l, • • • , n then linu..o 3v(x) =y^(x) uniformly
if
in
x,0gx^l.
We now consider (18). Suppose that ki, • • • , kq-i are points x,respectively
as near as possible to ki, ■ • ■ , kq-i. Then we determine
y(xi) so that at k[, • • ■ , k[-Y it satisfies the conditions imposed by
the theorem upon y(x) at ki, • ■ ■ , fe9_i. This entails the determination of yi(0), • • • , y»(0) as explained in the proof of the Theorem I.
The basic operation in each instance was Cramer's Rule for solving
linear algebraic equations.
In each case both numerator
and denominator
determinants
approach limits and the limit of the denominator is positive. These facts follow from the following considerations. Formula (10) shows that Avp(kj-i, k) as functions of k with
fixed p=n —pj and kj-i satisfy (1) where v = l, • • ■ , n. The initial
conditions
are A„(kj-i,
kj-i) = 1 and Arp(kj-i, &y_i)=0 when pj^v.
Consequently
when h approaches
0, Avp(kj_i, x/) approaches
a limit.
Denominator
determinants
namely iJ„(&y_i, kj, a, fi) have each element an A which approaches a limit. Consequently
Hv(kj-\, kj, a, fi)
itself approaches a limit. As a matter of fact Hv(kj-i, k, a, fi) as functions of k satisfy a set of equations (9) which are a precise analogue
of (1). Each of the ITs then has a limit which is a member of a solution of a set of linear differential equations with positive coefficients
and positive initial conditions.
It is then positive as remarked.
All
other operations
used in obtaining yi(0), ■ ■ ■ , yn(0) are a finite number of additions
and multiplications
of functions
which approach
limits. Hence yi(0), • • • , ^(O) approach
yi(0), • • • , y„(0) and the
corresponding
solution satisfies the conditions of the theorem,
as we have remarked k'j—^kj,j = l, ■ ■ • , q —l.
University
of South Carolina
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since